Construction of a neuroanatomical shape complex atlas from 3D MRI brain structures

Abstract

Brain atlas construction has attracted significant attention lately in the neuroimaging community due to its application to the characterization of neuroanatomical shape abnormalities associated with various neurodegenerative diseases or neuropsychiatric disorders. Existing shape atlas construction techniques usually focus on the analysis of a single anatomical structure in which the important inter-structural information is lost. This paper proposes a novel technique for constructing a neuroanatomical shape complex atlas based on an information geometry framework. A shape complex is a collection of neighboring shapes-for example, the thalamus, amygdala and the hippocampus circuit-which may exhibit changes in shape across multiple structures during the progression of a disease. In this paper, we represent the boundaries of the entire shape complex using the zero level set of a distance transform function S(x). We then re-derive the relationship between the stationary state wave function ψ(x) of the Schrödinger equation − 2 ∇ 2 ψ + ψ = 0 and the eikonal equation ∇S = 1 satisfied by any distance function. This leads to a one-to-one map (up to scale) between ψ(x) and S(x) via an explicit relationship. We further exploit this relationship by mapping ψ(x) to a unit hypersphere whose Riemannian structure is fully known, thus effectively turn ψ(x) into the square-root of a probability density * Corresponding author * * Principal corresponding author Email addresses: tichen@cise.ufl.edu (Ting Chen), anand@cise.ufl.edu (Anand Rangarajan), stephan.eisenschenk@neurology.ufl.edu (Stephan J. Eisenschenk), vemuri@cise.ufl.edu (Baba C. Vemuri) Preprint submitted to NeuroImageOctober 11, 2012 function. This allows us to make comparisons-using elegant, closed-form analytic expressions-between shape complexes represented as square-root densities. A shape complex atlas is constructed by computing the Karcher meanψ(x) in the space of square-root densities and then inversely mapping it back to the space of distance transforms in order to realize the atlas shape. We demonstrate the shape complex atlas computation technique via a set of experiments on a population of brain MRI scans including controls and epilepsy patients with either right anterior medial temporal or left anterior medial temporal lobectomies.

Distance Transform Gradient Density Estimation Using the Stationary Phase Approximation

“…By a similar argument, there can exist at most only a finite discrete set of θ for which R and τ κ3 ζ (3) (β, ω) respectively where both κ 2 and κ 3 ≥ 0.5 and ζ (2) and ζ (3) are some continuous bounded functions of β and ω. Hence, we can conclude that…”

“…As both r (1) k (β, ω) and r (2) k (β, ω) are also bounded functions, by the Lebesgue dominated convergence theorem, we get…”

“…where A 1 is the boundary along r(θ) = R (1) j (θ) and A 2 is the boundary along r(θ) = R (2) j (θ). Note that both A 1 and A 2 are composed of a finite sequence of straight line segments.…”

Revisioning the unification of syntax, semantics and statistics in shape analysis

“…In many cases, shape density functions are first estimated using Parzen windows or related methods. Subsequently, shape densities are matched using entropy minimization or other criteria to obtain shape deformation, shape atlases [17] and the like. Note the simplicity of the representation schemes which feature little to no explicit syntax.…”

Joint Morphometry of Fiber Tracts and Gray Matter Structures Using Double Diffeomorphisms

“…Most studies examine these objects independently focusing on a single anatomical structure [9][10][11][12][13]. Others propose multi-object analysis considering only a particular kind of mesh, either only sub-cortical structures [5][6][7][8] or only fiber-bundles [14][15][16]. However, an abnormal brain development due to a neuropsychiatric disorder can influence not only the shape of individual structures but also their organization.…”